Which of the following is CORRECT rationalised form of `(15)/(sqrt(5)+2)` ?

Options

A. A) `5sqrt(5)-6`
B. B) `5sqrt(5)-30`
C. C) `15sqrt(5)-30`
D. D) `45sqrt(5)-30`

Answer

To rationalize the expression \(\frac{15}{\sqrt{5} + 2}\), we will follow these steps:

### Step 1: Identify the conjugate
The conjugate of the denominator \(\sqrt{5} + 2\) is \(\sqrt{5} - 2\).

### Step 2: Multiply numerator and denominator by the conjugate
We multiply both the numerator and the denominator by the conjugate:

\[ \frac{15}{\sqrt{5} + 2} \cdot \frac{\sqrt{5} - 2}{\sqrt{5} - 2} \]

### Step 3: Simplify the denominator
Using the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \):

\[ (\sqrt{5})^2 - (2)^2 = 5 - 4 = 1 \]

So, the denominator simplifies to \(1\).

### Step 4: Simplify the numerator
Now, we simplify the numerator:

\[ 15(\sqrt{5} - 2) = 15\sqrt{5} - 30 \]

### Step 5: Write the final expression
Putting it all together, we have:

\[ \frac{15\sqrt{5} - 30}{1} = 15\sqrt{5} - 30 \]

Thus, the correct rationalized form of \(\frac{15}{\sqrt{5} + 2}\) is:

\[ 15\sqrt{5} - 30 \]